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Numerical simulation of unidimensional bubbly flow in linear and non-linear one parameter elastic liquid through a nozzles

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Abstract

This paper studies one dimensional bubbly cavitating flow of elastic fluids through micro sized nozzles of different shapes. Cavitating flows are applicable in wide range of applications in medical and engineering sciences, such as, the cleansing of teeth, ultrasound and cancer treatment. They are also responsible for erosion on metallic surfaces, damages to machinery, pumps etc. The above make advances in the field important and useful in reducing possible destructive effects of such flows. In current study two types of elastic fluid models namely neo-Hookean and linear elastic are considered. The nonlinear dynamics of bubbly mixture is modeled by incorporating the Rayleigh–Plesset equation. The system is modelled by nonlinear system of ordinary differential equations, which are reduced to non-dimensional form via suitable similarity transformations. The Runge–Kutta numerical technique of 4th order is utilized to solve the set of flow equations. The influences of various emerging parameters on bubble radius, velocity profile and pressure of bubble are illustrated graphically and discussed in detail.

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Abbreviations

A :

Cross sectional area of nozzle m\(^{2}\)

We:

Weber Number

\(\eta \) :

Bubble Population

u :

Velocity of the fluid

R :

Radius of the bubble

\(\rho _{l} \) :

Density of the fluid

\(\alpha \) :

Void fraction of the bubbly mixture

\(C_\mathrm{{p}}\) :

Fluid pressure coefficient

P :

Fluid pressure

t :

Time

\(\mu \) :

Dynamic viscosity of the fluid

\(p_\mathrm{{g}}\) :

Non-condensable gas inside the bubble

L :

Length of the nozzle

Re:

Reynolds number

S :

Surface tension

x :

Eulerian coordinates

\(\sigma \) :

Cavitation number

\(\gamma \) :

Dimensionless modulus of elasticity

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Faculty of Basic and Applied Sciences, International Islamic University, Islamabad, 44000, Pakistan

    Muhammad Shahid Nadeem & Ahmed Zeeshan

  2. Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

    F. Alzahrani

Authors
  1. Muhammad Shahid Nadeem
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  2. Ahmed Zeeshan
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  3. F. Alzahrani
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Corresponding author

Correspondence to Ahmed Zeeshan.

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The original online version of this article was revised: The author name Muhammad Shahid Nadeem was incorrectly written as Shahid Nadeem.

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Shahid Nadeem, M., Zeeshan, A. & Alzahrani, F. Numerical simulation of unidimensional bubbly flow in linear and non-linear one parameter elastic liquid through a nozzles. Eur. Phys. J. Spec. Top. 231, 571–581 (2022). https://doi.org/10.1140/epjs/s11734-022-00441-9

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